Double resonating beam force transducer with reduced longitudinal pumping

ABSTRACT

A double resonator beam force transducer configured to minimize longitudinal pumping by making the beams vibrate symmetrically. This can be accomplished by making the boundary conditions of the beams symmetrical or, if the boundary conditions are nonsymmetric, then by biasing the beams inwardly or outwardly to compensate for the nonsymmetrical boundary conditions of the beams. In the nonsymmetrical case where the beams would bow outwardly at their fundamental resonant frequency or an odd overtone thereof, an inward bias is provided to minimize longitudinal pumping. In the nonsymmetrical case where the beams would bow inwardly at an even overtone of the fundamental resonant frequency, an outward bias is provided to minimize longitudinal pumping. The inward or outward bias is provided in various embodiments by bowing the beams inwardly or outwardly, placing masses on the inner or outer edges of the beams, or tapering the inner or outer edges of the beams so that the width of the beams increase toward their midpoints.

DESCRIPTION

1. Technical Field

This invention relates to force transducers having double resonatingbeams extending between mounting pads, and more particularly, to a forcetransducer configured so that its beams vibrate inwardly and outwardlyon opposite sides of a straight mean position in order to minimize themagnitude of longitudinal forces applied to the mounting pads by thebeams.

2. Background Art

Double-beam resonators have been proposed for use as a force transducerto measure such physical properties as pressure, weight or acceleration.When used as a force transducer, a pair of substantially parallel beamsextend from respective mounting pads. Since the mounting pads formstationary nodes, the mounting pads may theoretically be connected to aforce-transmitting structure without the movement of the beams beingcoupled to such structure. Coupling of motion from the beams to theforce-transmitting structure would absorb energy from the beams and thusdegrade the quality factor or "Q" of the resonator.

One problem associated with double resonant beam force transducers thathas not be adequately recognized is the coupling of energy from thebeams to the force-transmitting structure because of longitudinalmovement of the mounting pads toward and away from each other resultingfrom deflection of the beams. Longitudinal movement or "pumping" occursbecause the distance between the mounting pads varies as the beamsdeflect from side to side. This longitudinal pumping is undesirable notonly for its degrading of the Q of the transducer, but also because itdegrades the linearity of the transducer. This nonlinearity arises whenthe resonant frequency of the beams at a certain force approaches theresonant frequency of the force-transmitting structure either alone orin combination with the mounting pads between which the beams extend. Asa result, when the response of the force transducer is linearized usingappropriate formulae, there is a residual error at certain values ofapplied force. The magnitude of the error depends upon the nature of theresonance in the surrounding structure, and it can range from relativelysmall values such as 2.5×10⁻⁵ to relatively large values such as2.5×10⁻³ of full scale.

If the resonance of the surrounding structure could be accuratelypredicted, the longitudinal pumping phenomena would not present aninsurmountable problem. This is because the surrounding supportstructure could be configured to have a resonance outside the resonantfrequency of the beams in their normal range of operation. However, theresonances in the support structure tend to be very complicated, becausethe structures are physically large compared to the dimensions of theforce transducer. For example, the double-beam force transducertypically vibrates at between 17 kHz and 40 kHz, depending on thespecific design. The fundamental resonance of the support structure istypically in the 1 kHz range. Thus, resonant frequencies of thesurrounding structure in the 17 kHz-40 kHz range are fairly highovertones of the fundamental, so that the mode spectrum of the supportstructure is very dense at the operating frequency of the resonator.These higher order resonant modes typically involve flexural, torsionaland extensional distortions which cannot be readily identified,controlled or predicted. Thus, it is impractical to design a supportstructure having a well-controlled mode spectrum at the operatingfrequency of the force transducer. The ideal solution would be to have asupport structure having no resonances over the entire operating rangeof the force transducer. However, since the operating frequency of aforce transducer with zero force frequency of 40 kHz typically variesfrom 36 kHz to 44 kHz as the force varies from full scale compression tofull scale tension, it is not possible to do so.

The degree of nonlinearity caused by longitudinal pumping is, to a largeextent, a function of the magnitude of the longitudinal pumping. Thus, areduction in the longitudinal pumping increases the linearity of therelationship between the resonant frequency of the beams and thecalculated applied force. It will be apparent that the movement of themounting pads in a longitudinal direction (i.e., toward and away fromeach other) is a function of the degree of lateral deflection of thebeams. Thus, when the beams are initially deflected laterally from astraight position, the degree of relative longitudinal movement of themounting pads is relatively slight. However, as the beams continue todeflect laterally, the rate at which the mounting pads movelongitudinally toward each other drastically increases. In short, therelative longitudinal movement of the mounting pads for a given lateraldeflection increases as the beams deflect laterally.

It has not heretofore been recognized that nonsymmetrical boundaryconditions at the junctions between the beams and the mounting padscauses the mean position of the beams to bow outwardly during vibrationat the fundamental resonant frequency of the beams or at odd overtonesthereof. Nor has it been recognized that these nonsymmetrical boundaryconditions cause the beams to bow inwardly during vibration at evenovertnes of the fundamental resonant frequency of the beams. As aresult, the beams do not bow inwardly and outwardly by equal amounts,but instead have a mean position to one side or the other. Therefore,for a given peak-to-peak lateral deflection of the beams, the maximumlateral position of the beams for nonsymmetrical end conditions issubstantially greater than for symmetrical conditions in which thedegree of inward and outward movement of the beams is equal. Thenonsymmetrical boundary conditions, by producing a relatively largelateral deflection, produce a relative large amount of longitudinalmovement or "pumping".

The phenomena of longitudinal motion imparted to a mounting pad has beenaddressed for tuning fork resonators used as time standards, such as inquartz watches. Thus, for example, the tines of such tuning forks can bemodified as shown in Tomikawa, et al., A Quartz Crystal Tuning Fork withModified Basewidth for a High Quality Factor: Finite Element Analysisand Experiments, IEEE Transactions on Sonics and Ultrasonics, Vol.SU-29, No. 6, July 1982; Tomikawa, et al., Second-Mode Tuning Forks forHigh Frequencies: Finite Element Analysis and Experiments, IEEETransactions on Sonics and Ultrasonics, Vol. SU-27, No. 5, September1980. However, the teachings of these references are not applicable todouble-beam resonators for several reasons. First, the beams of adouble-beam resonator deflect in a significantly different manner thando the tines of a tuning fork. A tuning fork tine deflects like afixed-free cantilever beam. In contrast, the beam of a double-beamresonator deflects like a fixed-fixed built-in beam. Each of these beamshas a different node pattern and resonating characteristic, thus eachgiven structural change affects each beam differently. Second, since thetines of a tuning fork are free at one end, the longitudinal motionimparted by the tines to the mounting pad is not a function ofvariations in the effective length of the tines as in the case of adoublebeam resonator. Instead, longitudinal motion is caused by inertia:i.e., the longitudinal component of the tine's movement. In effect, thetine's center of mass is moving toward and away from the mounting pad asit resonates laterally. In contrast, longitudinal pumping in double-beamresonators does not result from inertia. In fact, if the beams had zeromass, longitudinal pumping would still exist since the effective lengthof the beams would still vary.

DISCLOSURE OF THE INVENTION

It is an object of the invention to provide a double resonating beamforce transducer having improved force-induced frequency responsecharacteristics.

It is another object of the invention to provide a double resonatingbeam force transducer having a relatively high quality factor.

It is still another object of the invention to provide a doubleresonating beam force transducer that can be used with a supportstructure without the need to control spurious resonances in the supportstructure.

These and other objects of the invention are provided by a forcetransducer having a pair of parallel beams of substantially equal lengthextending between a pair of mounting pads which receive the forces to bemeasured. The force transducer is specially configured so that the beamsresonate equally inwardly and outwardly about a straight mean position.In one configuration of the force transducer, the beams are bowedinwardly toward each other in order to reduce longitudinal pumping forthe fundamental and odd overtones of the resonant frequency of thebeams. Conversely, the beams bow outwardly away from each other toreduce longitudinal pumping for even overtones of the resonant frequencyof the beams. In another configuration, the beams carry respectivemasses along a lateral edge at their midpoints, the inside edge beingused to reduce longitudinal pumping for fundamental and even overtonesof the resonant frequency of the beams and the outer edge being used toreduce longitudinal pumping for even numbered overtones of the resonantfrequency. In another configuration, the width of the beams is increasedtoward their midpoints, with the inside edges bowing inwardly forfundamental and odd overtones of the resonant frequency and the outeredges bowing outwardly for even overtones of the resonant frequency.

In still another embodiment of the inventive force transducer, theboundary conditions of the beams at their junction with the mountingpads are configured so that the boundary conditions are symmetrical sothat straight beams of uniform width can be used.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an isometric view of a typical double resonating bar forcetransducer.

FIGS. 2A and 2B are schematics illustrating the manner in whichlongitudinal forces are generated by the beams as a function of thelateral deflection of the beams.

FIGS. 3A and 3B are graphs plotting lateral displacement of the midpointof a beam and longitudinal displacement of the end of a beam as afunction of time for a curved mean lateral position of the beam and fora straight mean lateral position for a dual-beam force transducer havingone end fixed and the other end free.

FIG. 4 is a plan view of the junction between the beams of the forcetransducer and one mounting pad, provided for the purpose ofillustrating a major cause of longitudinal pumping.

FIG. 5 is a schematic representing the condition in which the resonatingbeams are coupled to a resonant support structure of the forcetransducer.

FIGS. 6A and 6B are graphs showing the frequency response of theillustrative system of FIG. 5 as a function of the resonant frequencydetermining spring constant of one resonant structure and a graph of theerror produced by coupling between the resonant structures shown in FIG.5.

FIGS. 7A and 7B are plan views of one embodiment of a configuration forcausing the beams to deflect inwardly and outwardly in equal amountsabout a straight mean position in which the beams are bowed inwardly oroutwardly.

FIGS. 8A and 8B are plan views; of another embodiment of a configurationfor causing the beams to deflect inwardly and outwardly in equal amountsabout a straight mean position in which the each beam carries a mass atits midpoint on either the inner or outer edge thereof.

FIGS. 9A and 9B are plan views. of another embodiment of configurationfor causing the beams to deflect inwardly and outwardly in equal amountsabout a straight mean position in which the widths of the beams increasetoward their midpoints by tapering either the inner or outer edges ofthe beams.

FIG. 10 is a plan view of another embodiment for causing the meanlateral positions of the beams to deflect inwardly and outwardly inequal amounts about a straight mean position by configuring the boundaryconditions of the beams so that they are symmetrical on either sides ofthe beams.

FIG. 11 is a graph showing the degree of longitudinal pumping as afunction of mounting pad geometry for operating frequencies at thefundamental, second and third overtones of the resonant frequencies ofthe beams.

FIGS. 12A and 12B are other embodiments for causing the beam to deflectinwardly and outwardly in equal amounts about a straight mean positionby making the boundary condit ions of the beams symmetrical.

BEST MODE FOR CARRYING OUT THE INVENTION

A double resonating beam force transducer, as illustrated in FIG. 1,includes two mounting pads 10,12 having a pair of generally parallelbeams 14,16 extending therebetween. The beams 14,16 are separated fromeach other by a slot 24. The force transducer is preferably formed froma piezoelectric material, such as quartz. Electrodes 28,30, in the formof films or coatings, extend onto the beams 14, 16, as shown in thedrawing. The electrodes 28,30 are connected to a conventional oscillator32 which applies an AC signal to the electrodes 28,30, which makes thebeams 14,16 vibrate inwardly and outwardly 180° out of phase from eachother. The frequency of the AC signal applied to the electrodes 28,30 isdetermined by the resonant frequency of the beams 10,12.

The mounting pads 10,12 are mounted on respective support structures50,52, which are typically significantly more massive than the mountingpads 10,12. Longitudinal forces (i.e., forces acting along thelongitudinal axis of the beams 14,16) are applied to the mounting pads10,12 through the support structures 50,52 to cause the resonantfrequency of the beams 14,16 to vary. The support structure 50,52 thusact as force-transmitting means. The resonant frequency increasesresponsive to tensional forces applied to the beams 14,16 and decreasesresponsive to compressive forces applied to the beams 14,16.

As mentioned above, the resonating beams 14,16 move laterally (i.e.,from side to side) at 180° from each other so that they aresimultaneously moving either inwardly or outwardly. As a result, therotational moments that the beams 14,16 apply to the mounting pads 10,12are theoretically equal and opposite each other. The mounting pads 10,12thus act as stationary nodes and, therefore, theoretically do not coupleenergy from the beams 14,16 to the support structure 50,52. It isimportant that energy not be coupled to the support structure, sinceenergy transferred from the beams 14,16 reduces the quality factor of"Q" of the force transducer with a resulting degradation in performance.

Although the mounting pads 10,12 acting as stationary nodes are fairlyeffective in preventing rotational moments and lateral forces from beingapplied to the support structure 50,52, they do not prevent all energytransfer from the beams 14,16 to the support structure 50,52. This isbecause deflection of the beams 14,16 causes the mounting pads 10,12 tobe pulled toward each other. Thus, lateral movement of the beams 14,16causes longitudinal forces to be applied between the mounting pads10,12. This "longitudinal pumping" causes energy to be transferred fromthe beams 14, 16 to the support structure 50,52 through the mountingpads 10,12. When the frequency of this longitudinal pumping approachesthe resonant frequency of higher order modes in the support structure50,52, the operating frequency of the force transducer 4 is pulled tosuch resonant frequency. This adversely affects the linearity of thefrequency versus force relationship, thereby degrading accuracy.

Although longitudinal pumping cannot be entirely eliminated, themagnitude of the longitudinal pumping depends to a large extent on thenature of the lateral movement of the beams 14,16. With reference toFIG. 2A, a beam 16 is shown schematically in three positions duringresonance. Position 16 is its innermost position, 16' is its outermostposition and 16" is the midpoint or mean position.

It will be apparent from 2A that the longitudinal position of the beammidpoint varies in disproportion to the lateral portion of the beam 16.Thus, the longitudinal position of the beam at 16' has movedsubstantially more than twice its position at 16" even though thelateral position at 16' is only twice that at 16". The degree oflongitudinal pumping for a beam deflecting inwardly and outwardly inequal amounts is substantially less, as illustrated in FIG. 2B. Thus, ifthe beams 14,16 can be made to deflect inwardly and outwardly insubstantially equal amounts so that the means or average position of thebeams 14,16 is straight, the magnitude of the longitudinal pumping canbe minimized.

The movement of the beams 14,16 for the nonsymmetrical and symmetricalconditions is plotted as a function of time in FIG. 3. In FIG. 3A, thebeams 14,16 are moving, as shown in FIG. 2A, from the straight or zerolateral position outwardly to a positive lateral position, as shown inthe top waveform of FIG. 3A. This nonsymmetric lateral movement causesthe ends of the beams to move longitudinally toward each other of thesame frequency as the operating frequency, as shown in the lowerwaveform of FIG. 3A. In contrast, for the same degree of symmetriclateral movement about a straight mean position shown in the top of FIG.3B, the magnitude of the longitudinal movement or pumping issubstantially less. Also, of course, the frequency of the longitudinalpumping in FIG. 3B is twice that of the operating frequency of the forcetransducer since the beams 14,16 move through the straight positiontwice each cycle.

The reason for the tendency of the beams 14,16 to resonatenonsymetrically, as illustrated in FIG. 2A, is the nonsymmetricalboundary conditions of the beams 14,16 at their junction with themounting pads 10,12. The interface between the beams 14,16 and mountingpad 10 is illustrated in greater detail in FIG. 4. It will be noted thatthe boundary conditions for the beam 16 adjacent the mounting pad 10 arenot symmetric in that no material is present in the region A-B whiletuning fork material is present in the region C-D on the opposite sideof the beam 16. As a result, the restoring force in the outwarddirection produced by the material in the region C-D is greater than therestoring force in the inward direction because of the absence of anymaterial in the region A-B. This asymmetry of boundary conditions causesthe average or mean position of the beams 14,16 to bow outwardly duringoperation at the fundamental resonant frequency of the beams 14,16 orodd overtones of the fundamental resonant frequency. Although it is notintuitively obvious, for even overtones of the fundamental resonantfrequency, the beams 14,16 swing inwardly further than they swingoutwardly so that the average or mean position of the beams 14,16 bowsinwardly.

As mentioned above, coupling between the beams 14,16 and the supportingstructure 50,52 resulting rom longitudinal pumping not only degrades theQ of the force transducer, but is also adversely affects the linearityof the frequency response as a function of applied force. This phenomenacan be better understood with reference to the schematic of FIG. 5. Asillustrated in the drawing, a first resonant structure consisting ofspring 60, mass 62 and damper 64 is coupled to a second resonatorstructure consisting of spring 66, mass 68 and damper 70 by spring 72.The frequency of resonance of each structure is given by the formula:

    F=(1/2π) (K/M).sup.1/2

For illustrative purposes, assume that the characteristics of the spring66, mass 68 and damper 70 of the structure on the right-hand side ofFIG. 5 are fixed. Also assume that the mass 62 and damper 64 on theleft-hand side are also fixed, but the spring constant of spring 60 isadjustable so that the resonant frequency of the resonant structure ofthe left-hand side of FIG. 5 is adjustable. As a result, the resonantfrequency of the structure on the left-hand side of FIG. 5 would vary ina smooth curve as the spring constant K varies. However, with thecoupling provided by spring 72, the resonant frequency of the structureon the left-hand is pulled toward the resonant frequency of thestructure on the right-hand side, as illustrated in the graph of FIG. 6Ain which the resonant frequency F₁ of the structure on the left and theresonant frequency F₂ of the structure on the right are plotted as afunction of the spring constant K of spring 60.

The resonant structure on the left-hand side of FIG. 5 represents thedouble resonant beam force transducer in which the spring constant Kvaries in accordance with the applied force. The resonant structure onthe right-hand side of FIG. 5 represents the support structure for theforce transucer in a simplistic form, since, in reality, the supportstructure has a large number of highly unpredictable and uncalculableresonances. The spring 72 represents the coupling between the beams14,16 and the support structure 50,52 resulting from longitudinalpumping of the force transducer. As a result, the frequency ofoscillation of the force transducer does not vary as a smooth functionof the applied force, but is instead pulled toward the various resonantfrequencies of the support structure. Each time this phenomenon occurs,a residual error occurs, such as shown in FIG. 6B for the simplifiedsystem of FIG. 5. The residual error has a region of positive and thennegative values as the force applied to the force transducer varies.

While the error shown in FIG. 6B cannot be entirely eliminated, it canbe greatly reduced by reducing either the degree of coupling between thebeams 14,16 and the support structures 50,52 or by reducing themagnitude of longitudinal pumping. As illustrated in FIGS. 2 and 3, thelongitudinal pumping can be minimized by configuring the beams 14,16 sothat they deflect substantially equally inwardly and outwardly.

One configuration for configuring the beams 14,16 to minimizelongitudinal pumping is illustrated in FIG. 7. In the embodiment of FIG.7, the beams 14,16 are initially curved in the opposite direction fromthe average curvature they would assume in operation because of thenonsymmetrical boundary conditions. Since the beams 14,16 resonate abouta mean position curving outwardly in the case of operation at thefundamental or odd overtones of the fundmental resonant frequency, thebeams 14,16 may be configured to minimize longitudinal pumping by bowingthe beams 14,16 inwardly, as illustrated in FIG. 7A. It has beencalculated and determined experimentally that the gap 24 between thebeams 14, 16 at the midpoint should be narrower than the width of thegap 24 at either end by about 2 percent to 3 percent of a width of abeam 14,16. The optimum configuration is for the midpoint of the gap 24to be narrower than the end of the gap 24 by about 2.5 percent of thewidth of a beam 14,16.

Insofar as the even overtones of the fundamental resonant frequencycause the beams 14,16 to bow inwardly, the congifuation of FIG. 7B, inwhich the beams 14,16 bow outwardly, can be used to minimizelongitudinal pumping at the even overtones of the fundamental resonantfrequency. The gap 24 at the midpoint should be wider than the gap 24 ateither end by approximately 11 percent to 15 percent of the width ofeach beam 14,16. The optimum configuration is for the gap 24 at themidpoint to be wider than the gap 24 at either end by about 13 percentof the width of the beams 14,16.

Another configuration for causing the beams 14,16 to resonatesymmetrically is shown in FIG. 8. In the embodiment of FIG. 8A, a lumpedmass 80,82 is formed on the inner edge of each beam 14,16 at theirmidpoints such that the centers of mass of the beams 14,16 are shiftedinwardly. By shifting the center of mass inwardly, the masses 80,82compensate for the tendency of the beams 14, 16 to bow outwardly whenthe force transducer is operating at the fundamental resonant frequencyor an odd numbred overtone of the resonant frequency. For even overtonesof the fundamental resonant frequency, masses 84,86 are formed on theouter edge of the beams 14,16 at their midpoints. The masses 84, 86shift the centers of mass of the beams 14,16 outwardly to compensate forthe inward bowing of the beams 14,16 at even overtones of thefundamental resonant frequency.

Although the embodiment of FIG. 8 utilizes discrete masses 80-86positioned on the edges of the beams 14,16, the masses may also bedistributed along the edges, as illustrated in FIG. 9. Thus, in theembodiment of FIG. 9A, the inner edges of the beams 14,16 are taperedinwardly, although the outer edges are straight. As a result, thecenters of mass of the beams are shifted inwardly to counteract thetendency of the beams 14,16 to bow outwardly at the fundamental resonantfrequency and odd overtones of the fundamental resonant frequency. Theincreased material along the inner edges of the beams, 14,16 also causesthe beams 14,16 to act somewhat like inwardly bowed beams, such asillustrated in FIG. 7A.

For even numbered overtones of the fundamental resonant frequency, themass may be distributed on the outer edges of the beams 14,16 tocorrespond to the masses 84,86 in FIG. 8B and somewhat assume the shapeof outwardly bowed beams, such as illustrated in FIG. 7B.

Rather than configure the beams 14,16 for minimum longitudinal pumpingby changing the shape or mass distribution of the beams, longitudinalpumping can be minimized by making the boundary conditions of the beamssymmetrical, such as illustrated in FIG. 10. Making the boundaryconditions at the ends symmetric causes the beams to swing equallyinwardly and outwardly for operation at the fundamental resonantfrequency as well as at any even or odd overtone thereof. In theembodiment illustrated in FIG. 10, a lateral projection or "outrigger"80 is provided to supply an inward restoring force that equalizes theoutward restoring force provided by the portions of the mounting pads10,12 beneath the gap 24. It can be shown that the optimum width W₀ ofthe outrigger 80 is approximately half the width of the gap 24. Themagnitude of the pumping motion at the resonant frequency for variouswidths W₀ of the outrigger 80 as a function of the width the gap 24 isillustrated in FIG. 11. It can be seen from FIG. 11 that the outriggerwidth W₀ of one-half of the w of the gap 24 is close to the optimumvalue for the second and third overtones of the resonant frequency aswell as the fundamental resonant frequency.

Although one configuration for making the boundary conditions of thebeams 14,16 fork symmetrical has been illustrated in FIG. 10, it will beunderstood that other configurations may be used, including, withoutlimitation, the embodiments in FIG. 12. As illustrated in FIG. 12A, theoutriggers 80 of FIG. 10 may be replaced by tapered supports 82 whichprovide an inward restoring force that is equal in magnitude to theoutward restoring force provided by the base mounting pads 10,12 beneaththe gap 24. In the embodiment of FIG. 13B, the beams 14,16 are notchedat 84 to reduce the outward restoring force imparted to the beams 14,16by the mounting pads 10,12.

It is thus seen that the effects of longitudinal pumping can beminimized utilizing any of the embodiments expalined above andillustrated in the drawings. As a result of this reduced longitudinalpumping, the quality factor "Q" and linearity of double resonating beamfork force transducers are greatly improved.

We claim:
 1. A double-beam resonator having reduced longitudinalpumping, comprising a pair of generally parallel, spaced-apart beams ofsubstantially equal length extending between a pair of supports andincluding means for causing said beams to resonate in opposite lateraldirections, said supports having a lateral dimension that is greaterthan the distance between the outer edges of said beams so that at leasta portion of said supports extend laterally beyond the outer edges ofsaid beams, said supports extending from respective mounting pads havinga lateral dimension that is greater than the lateral dimension of saidsupports such that said mounting pads extend laterally beyond the outeredges of said supports, said supports providing inward restoring forcesto said beams to cause said beams to resonate inwardly and outwardly insubstantially equal amounts about mean positions that are substantiallystraight, thereby minimizing the lateral deviation of said beams fromsaid mean position in order to minimize the longitudinal pumping of saidbeams.
 2. A double-beam resonator having reduced longitudinal pumping,comprising a pair of generally parallel beams of substantially equallength extending between a pair of mounting pads and including means forcausing said beams to resonate in opposite lateral directions, saidbeams bowing inwardly toward each other at their midpoints to cause saidbeams to resonate inwardly and outwardly in substantially equal amountsabout mean positions that are substantially straight, thereby minmizingthe lateral deviation of said beams from said mean position in order tominimize the longitudinal pumping of said beams.
 3. The resonator ofclaim 2 wherein the spacing between said beams at their midpoints isless than the spacing between said beams at their ends by about 2percent to 3 percent of the width of each beam.
 4. The resonator ofclaim 3 wherein the spacing between said beams at their midpoints isless than the spacing between said beams at their ends by about 2.5percent of the width of each beam.
 5. A double-beam resonator havingreduced longitudinal pumping, comprising a pair of generally parallelbeams of substantially equal length extending between a pair of mountingpads and including means for causing said beams to resonate in oppositelateral directions, said beams bowing outwardly away from each other attheir midpoints to cause said beams to resonate inwardly and outwardlyin substantially equal amounts about mean positions that aresubstantially straight, thereby minimizing the lateral deviation of saidbeams from said mean position in order to minimize the longitudinalpumping of said beams.
 6. The resonator of claim 5 wherein the spacingbetween said beams at their midpoints is greater than the spacingbetween said beams at their ends by about 11 percent to 15 percent ofthe width of each beam.
 7. The resonator of claim 6 wherein the spacingbetween said beams at their midpoints is greater than the spacingbetween said beams at their ends by about 13 percent of the width ofeach beam.
 8. A double-beam resonator having reduced longitudinalpumping, comprising a pair of generally parallel beams of substantiallyequal length extending between a pair of mounting pads and includingmeans for causing said beams to resonate in opposite lateral directions,said beams each carrying a mass at its midpoint along one lateral edgethereof to cause said beams to resonate inwardly and outwardly insubstantially equal amounts about mean positions that are substantiallystraight, thereby minimizing the lateral deviation of said beams fromsaid mean position in order to minimize the longitudinal pumping of saidbeams.
 9. The resonator of claim 8 wherein said masses are carried onthe inside edges of respective beams, thereby minimizing longitudinalpumping when said resonators operate at the fundamental resonantfrequency of said beams or an odd overtone thereof.
 10. The resonator ofclaim 8 wherein said masses are carried on the outside edges of saidrespective beams, thereby minimizing longitudinal pumping when saidresonators are operated at even overtones of the fundamental resonantfrequency of said beams.
 11. A double-beam resonator having reducedlongitudinal pumping, comprising a pair of generally parallel beams ofsubstantially equal length extending between a pair of mounting pads andincluding means for causing said beams to resonate in opposite lateraldirections, the width of each beam becoming greater toward the midpointthereof to cause said beams to resonate inwardly and outwardly insubstantially equal amounts about mean positions that are substantiallystraight, thereby minimizing the lateral deviation of said beams fromsaid mean position in order to minimize the longitudinal pumping of saidbeams.
 12. The resonator of claim 11 wherein each of said beams has aninside edge that tapers inwardly at its midpoint toward the other beamin order to minimize longitudinal pumping when said resonator isoperating at the fundamental resonant frequency of said beams or oddovertones thereof.
 13. The resonator of claim 11 wherein each of saidbeams has an outside edge that tapers outwardly at its midpoint awayfrom the other beam in order to minimize longitudinal pumping when saidresonator is operating at an even overtone of the fundamental resonantfrequency of said beams.
 14. A double-beam resonator having reducedlongitudinal pumping, comprising a pair of generally parallel beams ofsubstantially equal length extending between a pair of mounting pads andincluding means for causing said beams to resonate in opposite lateraldirections, each of said mounting pads extending laterally beyond theouter edges of said beams a distance approximately equal to half thelateral distance between said beams to increase the inward restoringforce exerted on said beams by said mounting pads to offset the outwardrestoring force exerted on said beams by said mounting pads, therebycausing said beams to resonate inwardly and outwardly in substantiallyequal amounts about mean positions that are substantially straight inorder to minimize the longitudinal pumping of said beams.
 15. Adouble-beam resonator having reduced longitudinal pumping, comprising apair of generally parallel beams of substantially equal length extendingbetween a pair of mounting pads and including means for causing saidbeams to resonate in opposite lateral directions, the outer edges ofsaid beams tapering outwardly as they join said mounting pads to provideinward restoring forces to said beams to offset the outward restoringforces applied to said beams by said mounting pads, thereby causing saidbeams to resonate inwardly and outwardly in substantially equal amountsabout mean positions that are substantially straight in order tominimize the longitudinal pumping of said beams.
 16. A double-beamresonator having reduced longitudinal pumping, comprising a pair ofgenerally parallel beams of substantially equal length extending betweena pair of mounting pads and including means for causing said beams toresonate in opposite lateral directions, the inside edge of each of saidbeams having formed therein a notch at each end thereof to reduce theoutward restoring force exerted on said beams by said mounting pads,thereby causing said beams to resonate inwardly and outwardly insubstantially equal amounts about mean positions that are substantiallystraight in order to minimize the longitudinal pumping of said beams.17. The resonator of claim 1, further including a pair of supportstructures on which respective mounting pads are mounted, said supportstructures being adapted to apply an externally generated force to saidbeams acting along the longitudinal axes of said beams so that saidexternal force alters the resonant frequency of said beams.
 18. A forcetransducer, comprising:a pair of spaced-apart mounting pads; a pair ofsupports extending from respective mounting pads, said supports havinglateral dimensions that are smaller than the lateral dimensions of saidmounting pads such that said mounting pads extend laterally beyond theedges of said supports; a pair of substantially parallel, spaced-apartbeams of substantially equal length extending between said supports,said supports having a lateral dimension that is greater than thedistance between the outer edges of said beams so that said supportsextend laterally beyond the outer edges of said beams, said supportsproviding inward restoring forces to said beams, thereby causing saidbeams to resonate inwardly and outwardly in substantially equal amountsso that they have mean positions that are substantially parallel to eachother, thereby minimizing the lateral deflection of said beams from saidmean position for a given magnitude of peak-to-peak lateral deflection;force-transmitting means for coupling an external force to said mountingpads acting along the longitudinal axes of said beams; and drive meansfor causing said beams to resonate in opposite directions at a frequencydetermined by the magnitude of the longitudinal force applied to saidbeams.
 19. A force transducer, comprising:a pair of spaced-apartmounting pads; a pair of substantially parallel beams of substantiallyequal length extending between said mounting pads, said beams bowinginwardly at their midpoints, thereby causing said beams to resonateinwardly and outwardly in substantially equal amounts when said beamsare resonating at their fundamental resonant frequency or an oddovertone thereof; force-transmitting means for coupling an externalforce to said mounting pads acting along the longitudinal axes of saidbeams; and drive means for causing said beams to resonate in oppositedirections at a frequency determined by the magnitude of thelongitudinal force applied to said beams.
 20. The force transducer ofclaim 19 wherein the spacing between said beams at their midpoints isless than the spacing between said beams at their ends by about twopercent to three percent of the width of each base.
 21. A forcetransducer, comprising:a pair of spaced-apart mounting pads; a pair ofsubstantially parallel beams of substantially equal length extendingbetween said mounting pads, said beams bowing outwardly away from eachother at their midpoints, thereby causing said beams to resonateinwardly and outwardly in substantially equal amounts when said beamsare resonating at even overtones of the fundamental resonant frequencyof said beams; force-transmitting means for coupling an external forceto said mounting pads acting along the longitudinal axes of said beams;and drive means for causing said beams to resonate in oppositedirections at a frequency determined by the magnitude of thelongitudinal force applied to said beams.
 22. The force transducer ofclaim 21 wherein the spacing between said beams at their midpoints isgreater than the spacing between said beams at their ends by about 11percent to 15 percent of the width of each beam.
 23. A force transducer,comprising:a pair of spaced-apart mounting pads; a pair of substantiallyparallel beams of substantially equal length extending between saidmounting pads, a mass mounted or each of said beams, said mass beingsymmetrically positioned at the midpoints of corresponding edges of saidbeams, said masses being mounted on the inside edges of said beams foroperation of said beams at the fundamental resonant frequency thereof oran odd overtone thereof, and on the outer edges of said beams foroperation at an even overtone of the fundamental resonant frequency ofsaid beams; force-transmitting means for coupling an external force tosaid mounting pads acting along the longitudinal axes of said beams; anddrive means for causing said beams to resonate in opposite directions ata frequency determined by the magnitude of the longitudinal forceapplied to said beams.
 24. The force transducer of claim 18 wherein themounting characteristics of said beams from said mounting pads arelaterally symmetrical so that the inward restoring force applied to saidbeams by said mounting pads is equal to the outward restoring forceapplied to said beams by said mounting pads.
 25. In a double resonatingbeam force transducer having a pair of beams connected between a pair ofmounting pads, a method of minimizing the longitudinal pumping producedby said beams, comprising biasing said beams in directions opposite tothe mean positions that said beams would otherwise assume whenresonating at the fundamental resonant frequency of said beams or anovertone thereof.
 26. The method of claim 25 wherein said beams arebiased inwardly to minimize longitudinal pumping of said beams when saidbeams are resonating at the fundamental resonant frequency of said beamsor an odd overtone thereof.
 27. The method of claim 25 wherein saidbeams are biased outwardly to minimize longitudinal pumping of saidbeams when said beams are operating at an even overtone of thefundamental resonant frequency of said beams.
 28. In a quartz crystalmicroresonator of the type including first and second stem portions, apair of integral tines extending between said stem portions, a slotbetween said tines, said slot having opposed crotches where said tinesjoin said stem portions, and a plurality of electrodes mounted on eachso said tines, spaced along the length thereof, said electrodes beingadapted for receipt of an AC voltage for causing said tines tooscillate, the improvement wherein each of said tines has a pair ofopposed shoulders where said tines join said stem portions and whereinthe widths of said shoulders at said opposite ends of said tines areapproximately equal and approximately equal to one-half of the width ofsaid crotches.
 29. The resonator of claim 1 wherein the side edges ofsaid supports taper inwardly from said mounting pads to the outer edgesof said beams.
 30. The resonator of claim 1 wherein said supports arerectangularly shaped.
 31. The resonator of claim 30 wherein the lateraldimensions of said rectangularly shaped supports are substantially equalto the sum of the distance between the outer edges of said beams and thedistance between the inner edges of said beams so that said supportsextend laterally beyond said beams by a distance of one-half thedistance between the inner edges of said beams.
 32. The resonator ofclaim 18 wherein the side edges of said supports taper inwardly fromsaid mounting pads to the outer edges of said beams.
 33. The resonatorof claim 18 wherein said supports are rectangularly shaped.
 34. Theresonator of claim 33 wherein the lateral dimensions of saidrectangularly shaped supports are substantially equal to the sum of thedistance between the outer edges of said beams and the distance betweenthe inner edges of said beams so that said supports extend laterallybeyond said beams by a distance of one-half the distance between theinner edges of said beams.
 35. A double-beam resonator having reducedlongitudinal pumping, comprising a pair of generally parallel,spaced-apart beams of substantially equal length extending between apair of mounting pads and including means for causing said beams toresonate in opposite lateral directions, said mounting pads havinglateral dimensions substantially equal to the sum of the lateraldistance between the outer edges of said beams and the lateral distancebetween the inner edges of said beams such that said mounting padsextend laterally beyond the edges of said beams by a distance ofapproximately one-half the lateral distance between the inner edges ofsaid beams.
 36. A double-beam resonator having reduced longitudinalpumping, comprising a pair of generally parallel, spacedapart beams ofsubstantially equal length extending between a pair of mounting pads andincluding means for causing said beams to resonate in opposite lateraldirections, said mounting pads having lateral dimensions that aregreater than the lateral distance between the outer edges of said beamsso that at least a portion of said mounting pads extends laterallybeyond the outer edges of said beams, the side edges of said mountingpads tapering inwardly to the outer edges of said beams.